Optimal. Leaf size=111 \[ \frac{4 b^3 (b d-a e)}{e^5 (d+e x)}-\frac{3 b^2 (b d-a e)^2}{e^5 (d+e x)^2}+\frac{4 b (b d-a e)^3}{3 e^5 (d+e x)^3}-\frac{(b d-a e)^4}{4 e^5 (d+e x)^4}+\frac{b^4 \log (d+e x)}{e^5} \]
[Out]
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Rubi [A] time = 0.197141, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{4 b^3 (b d-a e)}{e^5 (d+e x)}-\frac{3 b^2 (b d-a e)^2}{e^5 (d+e x)^2}+\frac{4 b (b d-a e)^3}{3 e^5 (d+e x)^3}-\frac{(b d-a e)^4}{4 e^5 (d+e x)^4}+\frac{b^4 \log (d+e x)}{e^5} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^5,x]
[Out]
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Rubi in Sympy [A] time = 58.6018, size = 100, normalized size = 0.9 \[ \frac{b^{4} \log{\left (d + e x \right )}}{e^{5}} - \frac{4 b^{3} \left (a e - b d\right )}{e^{5} \left (d + e x\right )} - \frac{3 b^{2} \left (a e - b d\right )^{2}}{e^{5} \left (d + e x\right )^{2}} - \frac{4 b \left (a e - b d\right )^{3}}{3 e^{5} \left (d + e x\right )^{3}} - \frac{\left (a e - b d\right )^{4}}{4 e^{5} \left (d + e x\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**5,x)
[Out]
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Mathematica [A] time = 0.118869, size = 119, normalized size = 1.07 \[ \frac{\frac{(b d-a e) \left (3 a^3 e^3+a^2 b e^2 (7 d+16 e x)+a b^2 e \left (13 d^2+40 d e x+36 e^2 x^2\right )+b^3 \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )}{(d+e x)^4}+12 b^4 \log (d+e x)}{12 e^5} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^5,x]
[Out]
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Maple [B] time = 0.011, size = 260, normalized size = 2.3 \[ -{\frac{4\,{a}^{3}b}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}+4\,{\frac{{a}^{2}d{b}^{2}}{{e}^{3} \left ( ex+d \right ) ^{3}}}-4\,{\frac{{d}^{2}a{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{4\,{d}^{3}{b}^{4}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}+{\frac{{b}^{4}\ln \left ( ex+d \right ) }{{e}^{5}}}-{\frac{{a}^{4}}{4\,e \left ( ex+d \right ) ^{4}}}+{\frac{{a}^{3}bd}{{e}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{3\,{d}^{2}{a}^{2}{b}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{4}}}+{\frac{{d}^{3}a{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{{b}^{4}{d}^{4}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}}-3\,{\frac{{a}^{2}{b}^{2}}{{e}^{3} \left ( ex+d \right ) ^{2}}}+6\,{\frac{ad{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}-3\,{\frac{{b}^{4}{d}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}-4\,{\frac{a{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}+4\,{\frac{d{b}^{4}}{{e}^{5} \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^5,x)
[Out]
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Maxima [A] time = 0.693789, size = 297, normalized size = 2.68 \[ \frac{25 \, b^{4} d^{4} - 12 \, a b^{3} d^{3} e - 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} + 48 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 36 \,{\left (3 \, b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 8 \,{\left (11 \, b^{4} d^{3} e - 6 \, a b^{3} d^{2} e^{2} - 3 \, a^{2} b^{2} d e^{3} - 2 \, a^{3} b e^{4}\right )} x}{12 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} + \frac{b^{4} \log \left (e x + d\right )}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2/(e*x + d)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.2017, size = 362, normalized size = 3.26 \[ \frac{25 \, b^{4} d^{4} - 12 \, a b^{3} d^{3} e - 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} + 48 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 36 \,{\left (3 \, b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 8 \,{\left (11 \, b^{4} d^{3} e - 6 \, a b^{3} d^{2} e^{2} - 3 \, a^{2} b^{2} d e^{3} - 2 \, a^{3} b e^{4}\right )} x + 12 \,{\left (b^{4} e^{4} x^{4} + 4 \, b^{4} d e^{3} x^{3} + 6 \, b^{4} d^{2} e^{2} x^{2} + 4 \, b^{4} d^{3} e x + b^{4} d^{4}\right )} \log \left (e x + d\right )}{12 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2/(e*x + d)^5,x, algorithm="fricas")
[Out]
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Sympy [A] time = 11.8826, size = 230, normalized size = 2.07 \[ \frac{b^{4} \log{\left (d + e x \right )}}{e^{5}} - \frac{3 a^{4} e^{4} + 4 a^{3} b d e^{3} + 6 a^{2} b^{2} d^{2} e^{2} + 12 a b^{3} d^{3} e - 25 b^{4} d^{4} + x^{3} \left (48 a b^{3} e^{4} - 48 b^{4} d e^{3}\right ) + x^{2} \left (36 a^{2} b^{2} e^{4} + 72 a b^{3} d e^{3} - 108 b^{4} d^{2} e^{2}\right ) + x \left (16 a^{3} b e^{4} + 24 a^{2} b^{2} d e^{3} + 48 a b^{3} d^{2} e^{2} - 88 b^{4} d^{3} e\right )}{12 d^{4} e^{5} + 48 d^{3} e^{6} x + 72 d^{2} e^{7} x^{2} + 48 d e^{8} x^{3} + 12 e^{9} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.217714, size = 377, normalized size = 3.4 \[ -b^{4} e^{\left (-5\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac{1}{12} \,{\left (\frac{48 \, b^{4} d e^{15}}{x e + d} - \frac{36 \, b^{4} d^{2} e^{15}}{{\left (x e + d\right )}^{2}} + \frac{16 \, b^{4} d^{3} e^{15}}{{\left (x e + d\right )}^{3}} - \frac{3 \, b^{4} d^{4} e^{15}}{{\left (x e + d\right )}^{4}} - \frac{48 \, a b^{3} e^{16}}{x e + d} + \frac{72 \, a b^{3} d e^{16}}{{\left (x e + d\right )}^{2}} - \frac{48 \, a b^{3} d^{2} e^{16}}{{\left (x e + d\right )}^{3}} + \frac{12 \, a b^{3} d^{3} e^{16}}{{\left (x e + d\right )}^{4}} - \frac{36 \, a^{2} b^{2} e^{17}}{{\left (x e + d\right )}^{2}} + \frac{48 \, a^{2} b^{2} d e^{17}}{{\left (x e + d\right )}^{3}} - \frac{18 \, a^{2} b^{2} d^{2} e^{17}}{{\left (x e + d\right )}^{4}} - \frac{16 \, a^{3} b e^{18}}{{\left (x e + d\right )}^{3}} + \frac{12 \, a^{3} b d e^{18}}{{\left (x e + d\right )}^{4}} - \frac{3 \, a^{4} e^{19}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-20\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2/(e*x + d)^5,x, algorithm="giac")
[Out]